Neural Computing and Applications

, Volume 29, Issue 10, pp 823–836 | Cite as

pth Moment synchronization of Markov switched neural networks driven by fractional Brownian noise

  • Xianghui Zhou
  • Jun Yang
  • Zhi Li
  • Dongbing Tong
Original Article


This paper deals with the pth moment synchronization problem for a type of the stochastic neural networks with Markov switched parameters and driven by fractional Brownian noise (FBNSNN). A method called time segmentation method, very different to the Lyapunov functional approach, has been presented to solve the above problem. Meanwhile, based on the trajectory of error system, associating with infinitesimal operator theory, we propose a sufficient condition of consensus for the drive–response system. The criterion of pth moment exponential stability for FBNSNN can guarantee the synchronization under the designed controller. Finally, two numerical examples and some illustrative figures are provided to show the feasibility and effectiveness for our theoretical results.


Synchronization pth Moment Time segmentation Switched neural networks Fractional Brownian noise 



This work is partially supported by the Open Research Fund Program of Institute of Applied Mathematics Yangtze University (Grant No. KF1602).


  1. 1.
    Zhang W, Li C, Huang T, Xiao M (2015) Synchronization of neural networks with stochastic perturbation via aperiodically intermittent control. Neural Netw 71:105–111 CrossRefGoogle Scholar
  2. 2.
    Wu Z, Shi P, Su H, Chu J (2014) Local synchronization of chaotic neural networks with sampled data and saturating actuators. IEEE Trans Cybern 44(12):2635–2645CrossRefGoogle Scholar
  3. 3.
    Ding S, Wang Z (2015) Stochastic exponential synchronization control of memristive neural networks with multiple time-varying delays. Neurocomputing 162:16–25CrossRefGoogle Scholar
  4. 4.
    Zhou X, Zhou W, Yang J (2015) A novel scheme for synchronization control of stochastic neural networks with multiple time-varying delays. Neurocomputing 159:50–57CrossRefGoogle Scholar
  5. 5.
    Wei Q, Liu D, Lewis FL (2015) Optimal distributed synchronization control for continuous-time heterogeneous multi-agent differential graphical games. Inf Sci 317:96–113CrossRefGoogle Scholar
  6. 6.
    Song Y, Wen S (2015) Synchronization control of stochastic memristor-based neural networks with mixed delays. Neurocomputing 156(25):121–128CrossRefGoogle Scholar
  7. 7.
    Zhou W, Zhu Q, Shi P, Su H, Fang Jianan, Zhou Liuwei (2014) Adaptive synchronization for neutral-type neural networks with stochastic perturbation and Markovian switching parameters. IEEE Trans Cybern 44(12):2848–2860CrossRefGoogle Scholar
  8. 8.
    Zhou X, Zhou W, Yang J (2015) Stochastic synchronization of neural networks with multiple time-varying delays and Markovian jump. J Frankl Inst 352(3):1265–1283MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Ma Y, Zheng Y (2015) Projective lag synchronization of Markovian jumping neural networks with mode-dependent mixed time-delays based on an integral sliding mode controller. Neurocomputing 168:626–636CrossRefGoogle Scholar
  10. 10.
    Tong D, Zhou W, Zhou X (2015) Exponential synchronization for stochastic neural networks with multi-delayed and Markovian switching via adaptive feedback control. Commun Nonlinear Sci Numer Simul 29:359–371MathSciNetCrossRefGoogle Scholar
  11. 11.
    Wu Z, Shi P, Su H, Chu J (2013) Stochastic synchronization of Markovian jump neural networks with time-varying delay using sampled data. IEEE Trans Cybern 43(6):1796–1806CrossRefGoogle Scholar
  12. 12.
    Yang J, Zhou W, Shi P, Yang X, Zhou Xianghui, Hongye Su (2015) Adaptive synchronization of delayed Markovian switching neural networks with Lévy noise. Neurocomputing 156(25):231–238CrossRefzbMATHGoogle Scholar
  13. 13.
    Wu Z, Park JH, Su H, Chu J (2012) Passivity analysis of Markov jump neural networks with mixed time-delays and piecewise-constant transition rates. Nonlinear Anal Real World Appl 13(5):2423–2431MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Mao X, Shen Y, Yuan C (2008) Almost surely asymptotic stability of neutral stochastic differential delay equations with Markovian switching. Stoch Process Appl 118:1385–1406MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Wang T, Zhao S, Zhou W, Yu W (2015) Finite-time state estimation for delayed Hopfield neural networks with Markovian jump. Neurocomputing 156(25):193–198CrossRefGoogle Scholar
  16. 16.
    Ksendal B (2005) Stochastic differential equations, 6th edn. Springer, BerlinGoogle Scholar
  17. 17.
    Mao X, Yuan C (2006) Stochastic differential equations with Markovian switching. Imperial College Press, LondonCrossRefzbMATHGoogle Scholar
  18. 18.
    Wang Z, Liu Y, Li M (2006) Stability analysis for stochastic Cohen–Grossberg neural networks with mixed time delays. IEEE Trans Neural Netw 17:814–820CrossRefGoogle Scholar
  19. 19.
    Zhou X, Zhou W, Dai A, Yang J (2014) Asymptotical stability of stochastic neural networks with multiple time-varying delays. Int J Control 88(3):613–621MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Chen H, Zhao Y (2015) Delay-dependent exponential stability for uncertain neutral stochastic neural networks with interval time-varying delay. Int J Syst Sci 46(14):2584–2597MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Wang Z, Shu H, Fang J (2006) Robust stability for stochastic Hopfield neural networks with time delays. Nonlinear Anal Real World Appl 7(5):1119–1128MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Yang L, Li Y (2015) Existence and exponential stability of periodic solution for stochastic Hopfield neural networks on time scales. Neurocomputing 167:543–550CrossRefGoogle Scholar
  23. 23.
    Peng J, Liu Z (2011) Stability analysis of stochastic reaction–diffusion delayed neural networks with Lévy noise. Neural Comput Appl 20(4):535–541CrossRefGoogle Scholar
  24. 24.
    Yang J, Zhou W, Shi P (2015) Synchronization of delayed neural networks with Lévy noise and Markovian switching via sampled data. Nonlinear Dyn 81(3):1179–1189CrossRefzbMATHGoogle Scholar
  25. 25.
    Zhou W, Yang J, Yang X (2014) Almost surely exponential stability of neural networks with Lévy noise and Markovian switching. Neurocomputing 145:154–159CrossRefGoogle Scholar
  26. 26.
    Caraballo T, Garrido-Atienza MJ, Taniguchi T (2011) The existence and exponential behavior of solutions to stochastic delay evolution equations with a fractional Brownian motion. Nonlinear Anal 74:3671–3684MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Diop MA, Garrido-Atienza MJ (2014) Retarded evolution systems driven by fractional Brownian motion with Hurst parameter \(H>\frac{1}{2}\). Nonlinear Anal Theory Methods Appl 97:15–29MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Duncan TE, Maslowski B, Pasik-Duncan B (2005) Stochastic equations in Hilbert space with a multiplicative fractional Gaussian noise. Stoch Process Their Appl 115:1357–1383MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Boufoussi B, Hajji S (2012) Neutral stochastic functional differential equations driven by a fractional Brownian motion in a Hilbert space. Stat Probab Lett 82:1549–1558MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Ke Y, Miao C (2015) Stability analysis of fractional-order Cohen–Grossberg neural networks with time delay. Int J Comput Math 92(6):1102–1113MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Zhang S, Yu Y, Wang H (2015) Mittag–Leffler stability of fractional-order Hopfield neural networks. Nonlinear Anal Hybrid Syst 16:104–121MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Wang H, Yu Y, Wen G (2015) Global stability analysis of fractional-order Hopfield neural networks with time delay. Neurocomputing 154:15–23CrossRefGoogle Scholar
  33. 33.
    Bao H, Cao J (2015) Projective synchronization of fractional-order memristor-based neural networks. Neural Netw 63:1–9CrossRefzbMATHGoogle Scholar
  34. 34.
    Rakkiyappan R, Cao J, Velmurugan G (2015) Existence and uniform stability analysis of fractional-order complex-valued neural networks with time delays. IEEE Trans Neural Netw Learn Syst 26(1):84–97MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Mishura Y (2008) Stochastic calculus for fractional Brownian motion and related processes. In: Lecture notes in mathematics, Springer, HeidelbergGoogle Scholar
  36. 36.
    Yu W, Cao J (2007) Synchronization control of stochastic delayed neural networks. Phys A 373:252–260CrossRefGoogle Scholar
  37. 37.
    Su S, Lin Z, Garcia A (2016) Distributed synchronization control of multiagent systems with unknown nonlinearities. IEEE Trans Cybern 46(1):325–338CrossRefGoogle Scholar

Copyright information

© The Natural Computing Applications Forum 2016

Authors and Affiliations

  • Xianghui Zhou
    • 1
  • Jun Yang
    • 2
  • Zhi Li
    • 1
  • Dongbing Tong
    • 3
  1. 1.School of Information and MathematicsYangtze UniversityJingzhouChina
  2. 2.School of Mathematics and StatisticsAnyang Normal UniversityAnyangChina
  3. 3.College of Electronic and Electrical EngineeringShanghai University of Engineering ScienceShanghaiChina

Personalised recommendations